Adjacent vertices of the all 0-1 programming polytope

RAIRO. R ECHERCHE OPÉRATIONNELLE

E ^GON B ^ALAS

M ^ANFRED W. P ^ADBERG

Adjacent vertices of the all 0-1 programming polytope

RAIRO. Recherche opérationnelle, tome 13, n^o1 (1979), p. 3-12

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R.A.I.R.O. Recherche opérationnelle/Opérations Research (vol. 13, n° 1, février 1979, p. 3 à 12)

ADJACENT VERTICES

OF THE ALL 0-1 PROGRAMMING POLYTOPE (*) (1) by Egon BALAS (2)

and Manfred W. PADBERG (3)

Abstract, — We give a constructive char ader ization ofadjacency relations between vertices of the convex huil offeasible 0-1 points of an all 0-1 program. This characterization can be used,for instance, to generate all vertices of the convex huil, adjacent to a given vertex. As a by-product, we establish a strong bound on the diameter of the convex huil offeasible 0-1 points.

Any linear 0-1 programming problem can be brought (by using binary expansion on the slack variables, when necessary, or other devices) to the form of an equality-constrained all 0-1 program:

(P) min { ex \ x e X, x integer } where

X = { x e Rn | Ax = b, x > 0 }

and where A is m x n, and Ax = b implies xj < 1, Vj e N = { ! , . . . , « } . We will assume, without loss of generality, that A has no identical columns or zero columns, and is of full row rank. Thej-th column of A will be denoted o,-.

Let (P') dénote the linear program associated with (P\ i. e., (P')

Further, let Xj be the convex huil of the feasible 0-1 points, i. e., Xl — conv { x e X \ x integer }

and let vert X (vert Xj) dénote the set of vertices of X (of Xj). X^s is the all 0-1 programming polytope referred to in the title.

(*) Manuscrit reçu Octobre 1977, révisé Janvier 1978.

(*) The research underlying this paper was supported by the National Science Foundation by grant MPS73-08534 A02,

(2) Carnegie Mellon University, Pittsburg.

O New York University.

R.Ai.R.O. Recherche opérationnelle/Opérations Research, 0399-0842/1979/3/$ 1.00

© Bordas-Dunod.

4 E. BALAS, M. W. PA0BERG

It is a well-known property of 0-1 programs that every feasible integer solution is basic. Hence, vert Xj ç vert X. Further, a solution associated with a feasible basis B, whose columns are indexée! by I, is integer if and only if ^at = b for some Q ç /, and Q is unique whenever it exists.

ieQ

Finally, if B is a feasible basis, / and J are the associated basic and nonbasic index sets, and ö, = B~1aj. To simplify notation, we assume the components of x to have been ordered so that I — { 1, . . . , m } ; thus the components of ïïj are aij9 i = 1, . . . , m. Observe that atj > 0 for at least one iel and every j e J, since X is bounded. Further, we dénote

* =(-;)

where es is the (n — m)-dimensional unit vector whose j-th component is 1 ; L e., the n-vector W is the j-th column of the Tucker-tableau. The fc-th compo- nent of W is denoted by ïï{.

Given a linear program, two bases are called adjacent if they differ in exactly one column. Two basic feasible solutions are called adjacent if they are adjacent vertices of the feasible set (i. e., distinct vertices contained in an edge, or 1-dimen- sional face). Two adjacent bases may be associated with the same solution;

while two adjacent basic feasible solutions may be associated with two bases that are not adjacent to each other.

The results of this paper were first shown in [1], [2] to hold for the set parti- tioning problem, a special case of the problem considered here. Most of the proofs given in [2] carry over to the gênerai case with only minor changes, but the main resuit (the sufficiency part of Theorem 3) requires a different approach. For the sake of completeness, we give all the proofs.

THEOREM 1 : Let x¹ and x² be two feasible integer solutions to (P')- Let B be a basis matrix associated with x\ let I and J be the index sets for the basic and nonbasic variables respectively, and for i = 1, 2, let

Q. = { j e N | x) = 1 } , Qf = N - Qi.

Then, denoting â} = B~iaj, jeJ,

1 keQ1nQ2

I % j = - 1 keQ2nQ1nI (2)

JeJ

"

0 ke(e

nô

)u(ê

n8

n/).

Proof: From the définition of Q,-, i = 1, 2, we have

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ADJACENT VERTICES OF THE ALL 0-1 PROGRAMMING POLYTOPE 5

which implies

Z flj= Z o * - Z ak

jeJ1nQ2 fceÖ2 keQml

= Z ak - Z ak

keQi keQ2nI

= Z _ ^ - Z

k

keQinQ2 fceQ2nÖin/

(by subtracting and adding £ ak).

keQmQ2

Premultiplying the last équation by B'¹ then produces (2), since the vee- tors ah keQx and kei, are columns of B, Q, E. D.

Next we state the converse of Theorem 1.

THEOREM 2 : Let xx be a feasible integer solution to (P'), let B, ƒ, J, 6 i a nd sj»

7 e J, be defined as in Theorem 1. Further, let the index set Q e J satisfy

0 or 1 ksQx

0 or - 1 ' - - ' " - {)

Then x2 defined by

2

' Y (4)

is a basic feasible solution to (P'),

j c ? = f i j e e2 = ö

J [ O otherwise where

Proof: Consider the problem (P') in (n + l)-space, obtained from (P') by augmenting A with the composite column au — Zaj * The transformed column 5^ = B~1au has an entry öfc^ = 1 for some ke Qly for otherwise (3) implies akh < 0, V/c G ƒ, which is impossible in view of the boundedness of the solution set. Pivoting on ïïkU = 1 yields a feasible solution x2 to (P')>

defined by

v 0 otherwise.

Since ^ v,

Z

; +

J* = Z

; =

jeS jeSuQ

it follows that x2 as defined by (5) is feasible for (F'). Since x2 is integer, it is also basic. From Theorem 1, relation (4) follows with Q = Jx n Q2. Ô. £. D.

A set g c J for which (3) holds will be called decomposable if it can be

vol. 13, n° 1, février 1979

6 E. BALAS, M. W. PADBERG

partitioned into two subsets, g* and g**, such that (3) remains true when g is replaced by Q* and g** respectively.

We now give a necessary and sufficient condition for two integer vertices of X to be adjacent on Xj.

THEOREM 3 : Let x1 and x2 be two feasible integer solutions to (P'\ with B, I, J, Qt and ö,-, je J, defined as in Theorem 1, and g2 = {j€ N \ xj = 1 } . Then x2 is adjacent to x1 on Xj (i. e.s x1 and x2 lie on an edgeof Xj) if and only if g = J n Q2 is not decomposable.

Proof : (i) Necessity. Suppose g is decomposable into g* and g**. Then the vectors

x< = *i _ £ « ' , f = 2, 3,4 (6)

J6S,

where S2 = g, S3 = g*, S4 = g**, and aJ' is defined by (1), are all feasible integer solutions to (P')> hence vertices of XT. Let nx = n0 be a supporting hyperplane for A^, such that TIX' = TC0 for î = 1, 2 and rcx < TI0, V x e X j . (If no such hyperplane exists, then x1 and x2 are not adjacent on XIy and the statement is proved.) Then from (6)

nx1 = nx2

= nx1 - w(J]ö/) = ÏI0

«ŒaO = 0, (7)

whereas

nx3 = Tix1 —

Tix4 = 71X1 - 7t( Y, âj) < K0 = nx1 JeQ**

or

n(£öO£0, 7i(Xâ^J)>0. (8)

Then from (7) and (8) we have

or TIX3 = nx4 = 7i0. Hence any supporting hyperplane for Xî that contains x1

and x2, also contains x3 and x4 ; i. e., x1 and x2 cannot lie on an edge of, or be adjacent on, Xj.

(H) Sufficiency. Suppose x1 and x2 are not adjacent on Xj. Let F be the face of minimal dimension of X^l9 which contains both x¹ and x² {F is clearly

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ADJACENT VERTIGES OF THE ALL 0-1 PROGRAMMING POLYTOPE 7

unique), and let x11, ..., xlp be the vertices of F adjacent to x1 on F. The (translated) convex polyhedral cone

C(x1)= { x l x - x1 + (xli-x1)kh\i>0,i = 1, . . . , p }

is known (see for instance [3]) to be the intersection of those halfspaces H+,

» = 1, . . . , p, such that x1 = p | H „ where Ht = bdHf. Since { H,+ }{ïf is a subset of the set of halfspaces whose intersection is Xu clearly Xt ç C(x1), and therefore every vertex x of F can be expressed as

Since x2 is not adjacent to x1, p > 2. From Theorems 1 and 2,

xu = x'- 2 > ' , i = l , . . . , p (10) and

where Qlf ç J, i = 1, . . . , p, and Ô — ^*

Since F is the lowest-dimensional face of Xx containing both x1 and x2, there exist Xt > 0 for i = 1, . . . , p, such that (9) holds with x = x2. For if not, then x2 is contained in a face F' of C(x^ such that

dim F' < dim C(xx) = dim F.

But i*1" = aff F' n Xj is a face of X7 that contains both x1 and x2 and dim F" = dim F' < dim F ,

which contradicts the assumption that F is the lowest-dimensional face of Xj containing both x1 and x2. Using (10) and the fact that (9) holds with x = x2

for some Xt > 0, i — 1, . . . , p, we have

x

= x

+ £

- x

)^

and from (11)

£ ! > ^with

P

which implies 6 = ^ J Qu.

i - l

vol. 13, n° 1, février 1979

8 E. BALAS, M. W. PADBERG

We now partition g into two subsets g* = gu and g** = g — g*. To complete the proof, we will show that (3) holds when g is replaced by g**

(for g* this follows from Theorem 1). This will be done by showing that x**

is a feasible integer solution to (P'\ where

x** = x

- £ ^

- x 2 +TL* (12)

JeQ*

Theorem 1 then implies that (3) holds with g replaced by g**.

First, from Theorem 1 and the définition of g*, x** is integer. Next we show by contradiction that x**^0. Suppose x?*<0. Then from (12), x2= 0 and

£ ct{= — 1 (since g* = g n ) . But from (10), this implies (lor i = 1) x,J =0, and hence _

1, ...,P) (13) But

xk2 = 4 - Z ( E aj[^, ^ > 0, i = 1, . . . , PI (14) hence xf > 05 contradicting our earlier finding that xf = 0. Hence, x** > 0.

Suppose on the other hand that xjf* > 1. By (12), x£ = 1 and £ ⣠= 1 (since g^{x l} = Q*). But from (10), this implies (for i = 1) that x£ = 1, and hence that (13) holds with reversed inequality. Again from (14) we conclude that xl < 1, contradicting our earlier finding that x£ = 1. Consequently, 0 < x^* < 1 for all keN. Finally, ^x** - b, since

where JR is the submatrix of A consisting of the columns aj9 jeJ, Hence x**

is a feasible 0-1 point. g. E. Dy

COROLLARY 3 . 1 : Let x1 and x2 be two vertices of Xj, and let B, /, J and â{

jeJ9 be defined as above. Then x2 is not adjacent to x1 on Xl9 if and only if there exists a family of p sets g1 (-^ J, i = 1, . . . , p, such that

(i) p>2;

(w) ö n n gl k= 05 Vi#fc;

(iw) the points

xi ^^xi _ Y^tSf^{9 i = h p}

JeQu

R.A.I.R.O. Recherche opérationnelle/Opérations Research

ADJACENT VERTICES OF THE ALL 0 - 1 PROGRAMMING POLYTOPE

are vertices of Xl9 adjacent to x1 ; and

(fc) x

= x

- j] I &

Proof: (oc) Necessity. If x1 and x2 are not adjacent on XT, then by Theorem 3 g = J n g2 can be partitioned into two subsets g* and g** such that (3) holds with g replaced by g* and g**. If

x* _ xi __ y ^j

and _.

are both adjacent to x1, the statement is proved; otherwise the reasoning can be applied to g* and/or g**, and can be repeated as many times as needed to obtain pairwise disjoint sets Q^lh i = 1, . . . , p , with p > 2, which are not decomposable.

p

(P) Sufficiency. If the condition holds, then g = KJQu = J n Ö 2 - Further- more, (7) is satisfied when g is replaced by Qu for i = 1, . . . , p.

From (HÏ) it follows that the vectors £ & a nd £ o7" are mutually ortho- gonal for all i ^ h, i, h e { 1, .. ., p } . Consequently, (3) also holds when g

p p

is replaced by \JQU- Thus g is decomposable into gx l and KjQu, hence x1 i = 2 i = 2

and x2 are not adjacent. g. E. D.

COROLLARY 3.2 : If x1 and x2 are two non-adjacent vertices of Xj related to each other by (iv\ then for any subset H of { 1, . . ., p },

Éeff

is a vertex of XI.

Proof: From (fif), the vectors £ ö⁷ and ]T â^J are pairwise orthogonal

p

u

i = 1

vol 13, n° 1, février 1979

p

for all ï, /Ï G {1, . .., p }, f 5e fc ; hence if (3) holds for Q = O Ôii> t n e n i l

also holds when g is replaced by KJQU. l~ n E D

10 E. BALAS, M. W. PADBERG

Corollary 3.2 can be given the following geometrie interprétation. A path on Xi between two vertices x, y is a séquence of vertices (x1, x2, . . . , xk), with x¹ = x, x^k = y, such that every pair of vertices x\ x^{t + 1}, i — 1, . . ., fe — 1, is connected by an edge of Xj ; the length of the path being k — 1. The edge- distance d{x, y) between x and )/ is the length of a shortest path between x and y. The diameter ô(Xj) of X7 is the longest edge-distance between any two vertices of Xx.

Let [a] dénote the largest integer less than or equal to the real number a.

For the next corollary, we assume that the matrix A defining Xt has no iden- tical columns.

COROLLARY 3 . 3 : 5(X7) < — where

L _l

n

z* = max { ]T Xj \ x e XT } .

Proof : Let x1, x2 be a pair of vertices of X7 which are at maximal edge- distance from each other, i. e., for which

Further, let B be a basis associated with x1 ; let I, J, (^ and ïï\ j e J, be defined as above.

From Corollary 3.1,

x

= x

- £ X ff (15)

and from Corollary 3.2, (15) holds with p > S ^ ) , since the séquence of vertices { x10, x11, . . . , x1^ } , of Xn where x10 = x1 and xlp = x2, with

defines a path of length p between x1 and x22. Now let P = { 1, . . . , p }, and let

Px = { ie P | £ 5£ = 1 for exactly one ke N } . If Pi = 0, then from (15) and the définition of 2*,

which, together with b(Xj) < p, proves the corollary. Suppose now that Px ^ 0.

Then for each iePl9 the vector £ W has at least two négative components.

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ADJACENT VERTICES OF THE ALL 0-1 PROGRAMMING POLYTOPE 11

For otherwise Qu is a singleton, say Qu = { h }, and et is of the form

(where é] is the k-dimensional unit vector whose j-th entry is 1); which implies that the nonbasic column ah of A is identical to a basic column, contrary to our assumption. Now let

where both x3 and x4 are vertices of Xj (Corollary 3.2). Then

*

= *

- I (- ZsO- I E*';

iePi iefii* ieP-Pi jeQu

but in view of

, Vi, heP, i

(16) implies that p < —— , where Q3 = {je N \ x] = 1 } . Hence, in view of 5(Xj) < p and | Q3 | < z*9 the corollary follows. Q. E. D.

REMARK; If in the définition of Xl9 A = (AG, Im) and b = (é71), where Im is the identity matrix of order m, e'" = (1, . . . , l j e T , and AG is the m x ( — incidence matrix of the complete undirected graph with m vertices, then

rz*~i

b(Xj) = — , since 5(X7) is achieved by the minimum distance between the empty matching and any maximum matching on the matching polytope. In this sense the upper bound on 5(Xj) given in the above Corollary is a strongest possible one.

The property stated in the next Theorem, which does not hold for arbitrary integer programs, has some interesting algorithmic implications.

THEOREM 4: Let x1 be a non-optimal vertex of Xj, let xll9 i = 1, . . . , fc, be those vertices of Xj adjacent to x1, and such that cxu < ex1, i = 1, . . . , L Then the convex polyhedral cone

C= { x | x = xx + ti(xli-x1)ki,Xi2:09i= 1, . . . , n }

contains an optimal vertex of Xj.

vol. 13, n° 1, février 1979

12 E. BALAS, M. W. PADBERG

Proof: Let x be an optimal vertex of Xx. If x is adjacent to x \ then xeC.

Otherwise, x can be expressed (Corollary 3.1) as

x = x

- £ E ^

= X¹ + f (X^U ~ X¹)

where x11, i = 1, . . . , p, are vertices of Xr adjacent to x1. Then 0 < ex1 - ex = £ £ câJ'

Let { 1, . . ., p } = Ps and let P+ = { 1, . . . , k }. Since ex < cx\ P+ # 0.

From Corollary 3.2, the point

is a vertex of Xu and from the définition of P+9

ex* = ex1 + £ c(xu - x1)

p

< ex1 -f £ c(xlf - x1) = ex.

Thus, since x is optimal, so is x*; and since the vertices xx\ ieP+ are among those that generate C5 clearly x* e C. Q. E. D.

The above results can be used to generate integer vertices of the feasible set X, adjacent to a given integer vertex x1. Namely, by systematically generat- ing composite columns of the form a^j* — £ ü{ where Q satisfies the require- ments for x1 — aj* to be a vertex of XT adjacent to x1, one can obtain all such vertices. The efficiency of a procedure based on these results will of course be highly dependent on the spécifie way in which they are used; and in view of the many options that are available, this topic requires further investigation.

REFERENCES

1. E. BALAS and M. PADBERG, On the Set Covering Problem. Opérations Research, 20, 1972, p. 1152-1161.

2. E. BALAS and M. PADBERG, On the Set Covering Problem. II: An Algorithm for Set Partitioning. Opérations Research, 23, 1975, p. 74-90.

3. J. STOER and C. WITZGALL, Convexity and Optimization in Finite Dimensions. I.

Springer, New York, 1970.

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